1. Field of the Invention
The present invention relates to Linear Dispersion Codes (LDCs). More particularly, the present invention relates to generating LDCs for use in Multiple-Input Multiple-Output (MIMO) wireless communication systems.
2. Description of the Related Art
In wireless MIMO communication systems, there is a trade-off between spectral efficiency and a signal's robustness. The trade-off is also referred to as a rate-diversity trade-off. In order to achieve a higher reliability, redundancy has to be added to an original signal, which results in a reduction of spectral efficiency. On the other hand, if a higher spectral efficiency is maintained, the signal becomes more vulnerable due to increased level of interference. Therefore, it is beneficial to find a good trade-off which ensures both high robustness and spectral efficiency.
The reference numerals [1]-[17] made in the below description correspond to the references listed at the end of the description, contents of each are hereby incorporated by reference.
Conventional Linear Dispersion Codes (LDCs) (see references [1] and [5]) have been proven to be able to provide a good rate-diversity trade-off. The concept of LDC in reference [1] provides a space-time coding framework. Based on specific criteria, LDC disperses a transmission signal across space and time (frequency) dimensions, exploiting both spatial and time (frequency) diversity. By design, the LDC subsumes a wide range of Space-Time Codes (STC), for example an Alamouti code in reference [2], Tarokh codes in reference [3], and a Vertical Bell Labs Layered Space-Time (V-BLAST) scheme in reference [4], also generally known as Spatial Multiplexing (SM).
The existing LDC designs mainly rely on one of the following methods:
Conventional STC designs, e.g., the Alamouti code in reference [2] and the Tarokh codes in reference [3];
Gradient-based search algorithms, e.g., Hassibi et al. in reference [1], Gohary et al. in reference [6], and Wang et al. in reference [7];
Frame theory, e.g., Heath et al. in reference [5]; and
Algebraic theory, e.g., the Diagonal Algebraic Space-Time (DAST) codes in references [8] and [9], the Threaded Algebraic Space-Time (TAST) codes in references [10] and [11], a golden code in reference [12], and perfect codes in reference [13].
Constructing an LDC is equivalent to generating a set of encoding matrices. Structure of the encoding matrices has a significant impact on the code's achievable performance.
Conventional STCs, e.g., the Alamouti code in reference [2] and the Tarokh codes in reference [3], which can be perfectly represented in the LDC form, were created following a principle of orthogonal design. However, it has been shown that non-orthogonal codes generally outperform orthogonal codes in high-rate scenarios in references [1], [5] and [6].
For non-orthogonal LDCs, one can resort to frame theory as suggested by Heath et al. in reference [5], where projection-based and unitary-based parameterization algorithms are proposed. A first approach adopts QR-decomposition to generate unitary dispersion matrices. However, the result of QR-decomposition is not unique and is not distributed with Haar measure in reference [14]. The second approach in reference [5] exploits Householder reflections in reference [15] to produce candidate unitary matrices, though some implementation issues for the determination of Householder matrices need to be considered, for instance an improper choice of a sign can result in numerical instability in reference [15].
Alternatively, Givens rotations in reference [15] can also be used in reference [6]. However, Givens rotations are less numerically stable than Householder reflections. Further, Givens rotations can only generate a constrained set of unitary matrices, which slightly limits search space.
Another possibility is to exploit algebraic theory, which led to the present invention of the DAST codes in references [8] and [9], the TAST codes in references [10] and [11], and the perfect codes in reference [13]. However, the design of these codes has resulted in some constraints. For example, DAST codes may outperform orthogonal codes in high dimensions only, such as with four transmit antennas in reference [8]. Perfect codes were found only in specific scenarios, namely for dimensions of 2×2, 3×3, 4×4, and 6×6, respectively, where dimension 2×2 is representative of the Golden Code in reference [12].
Known criteria for LDC optimization include ergodic channel capacity or Mutual Information (MI) in reference [1], Pair-wise Error Probability (PEP) in reference [3], Block Error Probability (BLEP) in reference [7], and the like. Due to a non-convex nature of cost functions, numerical methods, such as, for example, gradient-based search algorithms in references [1], [6] and [7], have been proved to be more convenient and effective.
The LDC is defined by a set of dispersion matrices which have a large number of entries. Depending on the optimization criterion adopted, theoretical analysis may be possible. However, in most of the known cases, numerical methods are proved to be more effective for finding good LDCs.
Certain known methods employ gradient-based methods, either classical ones or their variations, to conduct a search for LDCs. For example, a stochastic gradient algorithm was used in reference [7], which however can only converge to a local minimum.
As mentioned above, LDCs in references [1] and [5] have been proven to be able to provide a good rate-diversity trade-off. Depending on the design method, the structure and performance of LDC can vary significantly. A new way of constructing LDCs provides a convenient way to transform a set of predefined parameter vectors to a set of linear dispersion matrices, which fully define an LDC.
In a random unitary matrix theory, certain design methods are capable of exploring unconstrained and full space of unitary matrices, where new LDCs, superior to existing solutions, can be found.
Depending on practical application scenarios, the dimensions of the new LDCs generated can be arbitrarily square or non-square, thus providing a high flexibility for design and application.
Design methods may also be conveniently combined with any appropriate numerical optimization algorithms to search for new codes.
In contrast, unitary LDCs have been shown to be asymptotic optimal in reference [6]. By construction, certain new design methods exploring an entire space of unitary matrices according to the unique Haar measure, where optimized unitary LDCs with better performance than their existing counterparts can be found. Moreover, by using the design framework, the production of LDCs with flexible dimensions for both square and non-square cases can be facilitated.
Depending on the optimization method (used to select an LDC), the performance of the LDC can vary significantly. A new way of searching for optimized LDCs (i.e., a new way of determining an optimum LDC) may be introduced, which exploit a specifically designed and tuned Genetic Algorithm (GA). Genetic Algorithms (GAs), on the other hand, provide a power global optimization method that is efficient for complicated non-linear problems.
Furthermore, taking into account the constraints of the LDC optimization problem, the GA adopted in certain optimization frameworks is based on a real matrix encoding. Compared with the widely-used binary encoding, real encoding has the benefit of providing more degree of freedom in terms of increasing search granularity.
Moreover, by exploiting a unitary matrix transformation mechanism, the customized GA-based framework in certain embodiments can provide a full search for the unitary matrix space, where better LDCs can be found.
Therefore, a need exists for a method for generating LDCs for use in Multiple-Input Multiple-Output (MIMO) wireless communication systems.